LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
Loading...
Searching...
No Matches

◆ stgsen()

subroutine stgsen ( integer  IJOB,
logical  WANTQ,
logical  WANTZ,
logical, dimension( * )  SELECT,
integer  N,
real, dimension( lda, * )  A,
integer  LDA,
real, dimension( ldb, * )  B,
integer  LDB,
real, dimension( * )  ALPHAR,
real, dimension( * )  ALPHAI,
real, dimension( * )  BETA,
real, dimension( ldq, * )  Q,
integer  LDQ,
real, dimension( ldz, * )  Z,
integer  LDZ,
integer  M,
real  PL,
real  PR,
real, dimension( * )  DIF,
real, dimension( * )  WORK,
integer  LWORK,
integer, dimension( * )  IWORK,
integer  LIWORK,
integer  INFO 
)

STGSEN

Download STGSEN + dependencies [TGZ] [ZIP] [TXT]

Purpose:
 STGSEN reorders the generalized real Schur decomposition of a real
 matrix pair (A, B) (in terms of an orthonormal equivalence trans-
 formation Q**T * (A, B) * Z), so that a selected cluster of eigenvalues
 appears in the leading diagonal blocks of the upper quasi-triangular
 matrix A and the upper triangular B. The leading columns of Q and
 Z form orthonormal bases of the corresponding left and right eigen-
 spaces (deflating subspaces). (A, B) must be in generalized real
 Schur canonical form (as returned by SGGES), i.e. A is block upper
 triangular with 1-by-1 and 2-by-2 diagonal blocks. B is upper
 triangular.

 STGSEN also computes the generalized eigenvalues

             w(j) = (ALPHAR(j) + i*ALPHAI(j))/BETA(j)

 of the reordered matrix pair (A, B).

 Optionally, STGSEN computes the estimates of reciprocal condition
 numbers for eigenvalues and eigenspaces. These are Difu[(A11,B11),
 (A22,B22)] and Difl[(A11,B11), (A22,B22)], i.e. the separation(s)
 between the matrix pairs (A11, B11) and (A22,B22) that correspond to
 the selected cluster and the eigenvalues outside the cluster, resp.,
 and norms of "projections" onto left and right eigenspaces w.r.t.
 the selected cluster in the (1,1)-block.
Parameters
[in]IJOB
          IJOB is INTEGER
          Specifies whether condition numbers are required for the
          cluster of eigenvalues (PL and PR) or the deflating subspaces
          (Difu and Difl):
           =0: Only reorder w.r.t. SELECT. No extras.
           =1: Reciprocal of norms of "projections" onto left and right
               eigenspaces w.r.t. the selected cluster (PL and PR).
           =2: Upper bounds on Difu and Difl. F-norm-based estimate
               (DIF(1:2)).
           =3: Estimate of Difu and Difl. 1-norm-based estimate
               (DIF(1:2)).
               About 5 times as expensive as IJOB = 2.
           =4: Compute PL, PR and DIF (i.e. 0, 1 and 2 above): Economic
               version to get it all.
           =5: Compute PL, PR and DIF (i.e. 0, 1 and 3 above)
[in]WANTQ
          WANTQ is LOGICAL
          .TRUE. : update the left transformation matrix Q;
          .FALSE.: do not update Q.
[in]WANTZ
          WANTZ is LOGICAL
          .TRUE. : update the right transformation matrix Z;
          .FALSE.: do not update Z.
[in]SELECT
          SELECT is LOGICAL array, dimension (N)
          SELECT specifies the eigenvalues in the selected cluster.
          To select a real eigenvalue w(j), SELECT(j) must be set to
          .TRUE.. To select a complex conjugate pair of eigenvalues
          w(j) and w(j+1), corresponding to a 2-by-2 diagonal block,
          either SELECT(j) or SELECT(j+1) or both must be set to
          .TRUE.; a complex conjugate pair of eigenvalues must be
          either both included in the cluster or both excluded.
[in]N
          N is INTEGER
          The order of the matrices A and B. N >= 0.
[in,out]A
          A is REAL array, dimension(LDA,N)
          On entry, the upper quasi-triangular matrix A, with (A, B) in
          generalized real Schur canonical form.
          On exit, A is overwritten by the reordered matrix A.
[in]LDA
          LDA is INTEGER
          The leading dimension of the array A. LDA >= max(1,N).
[in,out]B
          B is REAL array, dimension(LDB,N)
          On entry, the upper triangular matrix B, with (A, B) in
          generalized real Schur canonical form.
          On exit, B is overwritten by the reordered matrix B.
[in]LDB
          LDB is INTEGER
          The leading dimension of the array B. LDB >= max(1,N).
[out]ALPHAR
          ALPHAR is REAL array, dimension (N)
[out]ALPHAI
          ALPHAI is REAL array, dimension (N)
[out]BETA
          BETA is REAL array, dimension (N)

          On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
          be the generalized eigenvalues.  ALPHAR(j) + ALPHAI(j)*i
          and BETA(j),j=1,...,N  are the diagonals of the complex Schur
          form (S,T) that would result if the 2-by-2 diagonal blocks of
          the real generalized Schur form of (A,B) were further reduced
          to triangular form using complex unitary transformations.
          If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
          positive, then the j-th and (j+1)-st eigenvalues are a
          complex conjugate pair, with ALPHAI(j+1) negative.
[in,out]Q
          Q is REAL array, dimension (LDQ,N)
          On entry, if WANTQ = .TRUE., Q is an N-by-N matrix.
          On exit, Q has been postmultiplied by the left orthogonal
          transformation matrix which reorder (A, B); The leading M
          columns of Q form orthonormal bases for the specified pair of
          left eigenspaces (deflating subspaces).
          If WANTQ = .FALSE., Q is not referenced.
[in]LDQ
          LDQ is INTEGER
          The leading dimension of the array Q.  LDQ >= 1;
          and if WANTQ = .TRUE., LDQ >= N.
[in,out]Z
          Z is REAL array, dimension (LDZ,N)
          On entry, if WANTZ = .TRUE., Z is an N-by-N matrix.
          On exit, Z has been postmultiplied by the left orthogonal
          transformation matrix which reorder (A, B); The leading M
          columns of Z form orthonormal bases for the specified pair of
          left eigenspaces (deflating subspaces).
          If WANTZ = .FALSE., Z is not referenced.
[in]LDZ
          LDZ is INTEGER
          The leading dimension of the array Z. LDZ >= 1;
          If WANTZ = .TRUE., LDZ >= N.
[out]M
          M is INTEGER
          The dimension of the specified pair of left and right eigen-
          spaces (deflating subspaces). 0 <= M <= N.
[out]PL
          PL is REAL
[out]PR
          PR is REAL

          If IJOB = 1, 4 or 5, PL, PR are lower bounds on the
          reciprocal of the norm of "projections" onto left and right
          eigenspaces with respect to the selected cluster.
          0 < PL, PR <= 1.
          If M = 0 or M = N, PL = PR  = 1.
          If IJOB = 0, 2 or 3, PL and PR are not referenced.
[out]DIF
          DIF is REAL array, dimension (2).
          If IJOB >= 2, DIF(1:2) store the estimates of Difu and Difl.
          If IJOB = 2 or 4, DIF(1:2) are F-norm-based upper bounds on
          Difu and Difl. If IJOB = 3 or 5, DIF(1:2) are 1-norm-based
          estimates of Difu and Difl.
          If M = 0 or N, DIF(1:2) = F-norm([A, B]).
          If IJOB = 0 or 1, DIF is not referenced.
[out]WORK
          WORK is REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
[in]LWORK
          LWORK is INTEGER
          The dimension of the array WORK. LWORK >=  4*N+16.
          If IJOB = 1, 2 or 4, LWORK >= MAX(4*N+16, 2*M*(N-M)).
          If IJOB = 3 or 5, LWORK >= MAX(4*N+16, 4*M*(N-M)).

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.
[out]IWORK
          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
[in]LIWORK
          LIWORK is INTEGER
          The dimension of the array IWORK. LIWORK >= 1.
          If IJOB = 1, 2 or 4, LIWORK >=  N+6.
          If IJOB = 3 or 5, LIWORK >= MAX(2*M*(N-M), N+6).

          If LIWORK = -1, then a workspace query is assumed; the
          routine only calculates the optimal size of the IWORK array,
          returns this value as the first entry of the IWORK array, and
          no error message related to LIWORK is issued by XERBLA.
[out]INFO
          INFO is INTEGER
            =0: Successful exit.
            <0: If INFO = -i, the i-th argument had an illegal value.
            =1: Reordering of (A, B) failed because the transformed
                matrix pair (A, B) would be too far from generalized
                Schur form; the problem is very ill-conditioned.
                (A, B) may have been partially reordered.
                If requested, 0 is returned in DIF(*), PL and PR.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
  STGSEN first collects the selected eigenvalues by computing
  orthogonal U and W that move them to the top left corner of (A, B).
  In other words, the selected eigenvalues are the eigenvalues of
  (A11, B11) in:

              U**T*(A, B)*W = (A11 A12) (B11 B12) n1
                              ( 0  A22),( 0  B22) n2
                                n1  n2    n1  n2

  where N = n1+n2 and U**T means the transpose of U. The first n1 columns
  of U and W span the specified pair of left and right eigenspaces
  (deflating subspaces) of (A, B).

  If (A, B) has been obtained from the generalized real Schur
  decomposition of a matrix pair (C, D) = Q*(A, B)*Z**T, then the
  reordered generalized real Schur form of (C, D) is given by

           (C, D) = (Q*U)*(U**T*(A, B)*W)*(Z*W)**T,

  and the first n1 columns of Q*U and Z*W span the corresponding
  deflating subspaces of (C, D) (Q and Z store Q*U and Z*W, resp.).

  Note that if the selected eigenvalue is sufficiently ill-conditioned,
  then its value may differ significantly from its value before
  reordering.

  The reciprocal condition numbers of the left and right eigenspaces
  spanned by the first n1 columns of U and W (or Q*U and Z*W) may
  be returned in DIF(1:2), corresponding to Difu and Difl, resp.

  The Difu and Difl are defined as:

       Difu[(A11, B11), (A22, B22)] = sigma-min( Zu )
  and
       Difl[(A11, B11), (A22, B22)] = Difu[(A22, B22), (A11, B11)],

  where sigma-min(Zu) is the smallest singular value of the
  (2*n1*n2)-by-(2*n1*n2) matrix

       Zu = [ kron(In2, A11)  -kron(A22**T, In1) ]
            [ kron(In2, B11)  -kron(B22**T, In1) ].

  Here, Inx is the identity matrix of size nx and A22**T is the
  transpose of A22. kron(X, Y) is the Kronecker product between
  the matrices X and Y.

  When DIF(2) is small, small changes in (A, B) can cause large changes
  in the deflating subspace. An approximate (asymptotic) bound on the
  maximum angular error in the computed deflating subspaces is

       EPS * norm((A, B)) / DIF(2),

  where EPS is the machine precision.

  The reciprocal norm of the projectors on the left and right
  eigenspaces associated with (A11, B11) may be returned in PL and PR.
  They are computed as follows. First we compute L and R so that
  P*(A, B)*Q is block diagonal, where

       P = ( I -L ) n1           Q = ( I R ) n1
           ( 0  I ) n2    and        ( 0 I ) n2
             n1 n2                    n1 n2

  and (L, R) is the solution to the generalized Sylvester equation

       A11*R - L*A22 = -A12
       B11*R - L*B22 = -B12

  Then PL = (F-norm(L)**2+1)**(-1/2) and PR = (F-norm(R)**2+1)**(-1/2).
  An approximate (asymptotic) bound on the average absolute error of
  the selected eigenvalues is

       EPS * norm((A, B)) / PL.

  There are also global error bounds which valid for perturbations up
  to a certain restriction:  A lower bound (x) on the smallest
  F-norm(E,F) for which an eigenvalue of (A11, B11) may move and
  coalesce with an eigenvalue of (A22, B22) under perturbation (E,F),
  (i.e. (A + E, B + F), is

   x = min(Difu,Difl)/((1/(PL*PL)+1/(PR*PR))**(1/2)+2*max(1/PL,1/PR)).

  An approximate bound on x can be computed from DIF(1:2), PL and PR.

  If y = ( F-norm(E,F) / x) <= 1, the angles between the perturbed
  (L', R') and unperturbed (L, R) left and right deflating subspaces
  associated with the selected cluster in the (1,1)-blocks can be
  bounded as

   max-angle(L, L') <= arctan( y * PL / (1 - y * (1 - PL * PL)**(1/2))
   max-angle(R, R') <= arctan( y * PR / (1 - y * (1 - PR * PR)**(1/2))

  See LAPACK User's Guide section 4.11 or the following references
  for more information.

  Note that if the default method for computing the Frobenius-norm-
  based estimate DIF is not wanted (see SLATDF), then the parameter
  IDIFJB (see below) should be changed from 3 to 4 (routine SLATDF
  (IJOB = 2 will be used)). See STGSYL for more details.
Contributors:
Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.
References:
  [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
      Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
      M.S. Moonen et al (eds), Linear Algebra for Large Scale and
      Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.

  [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified
      Eigenvalues of a Regular Matrix Pair (A, B) and Condition
      Estimation: Theory, Algorithms and Software,
      Report UMINF - 94.04, Department of Computing Science, Umea
      University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working
      Note 87. To appear in Numerical Algorithms, 1996.

  [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
      for Solving the Generalized Sylvester Equation and Estimating the
      Separation between Regular Matrix Pairs, Report UMINF - 93.23,
      Department of Computing Science, Umea University, S-901 87 Umea,
      Sweden, December 1993, Revised April 1994, Also as LAPACK Working
      Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1,
      1996.

Definition at line 448 of file stgsen.f.

451*
452* -- LAPACK computational routine --
453* -- LAPACK is a software package provided by Univ. of Tennessee, --
454* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
455*
456* .. Scalar Arguments ..
457 LOGICAL WANTQ, WANTZ
458 INTEGER IJOB, INFO, LDA, LDB, LDQ, LDZ, LIWORK, LWORK,
459 $ M, N
460 REAL PL, PR
461* ..
462* .. Array Arguments ..
463 LOGICAL SELECT( * )
464 INTEGER IWORK( * )
465 REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
466 $ B( LDB, * ), BETA( * ), DIF( * ), Q( LDQ, * ),
467 $ WORK( * ), Z( LDZ, * )
468* ..
469*
470* =====================================================================
471*
472* .. Parameters ..
473 INTEGER IDIFJB
474 parameter( idifjb = 3 )
475 REAL ZERO, ONE
476 parameter( zero = 0.0e+0, one = 1.0e+0 )
477* ..
478* .. Local Scalars ..
479 LOGICAL LQUERY, PAIR, SWAP, WANTD, WANTD1, WANTD2,
480 $ WANTP
481 INTEGER I, IERR, IJB, K, KASE, KK, KS, LIWMIN, LWMIN,
482 $ MN2, N1, N2
483 REAL DSCALE, DSUM, EPS, RDSCAL, SMLNUM
484* ..
485* .. Local Arrays ..
486 INTEGER ISAVE( 3 )
487* ..
488* .. External Subroutines ..
489 EXTERNAL slacn2, slacpy, slag2, slassq, stgexc, stgsyl,
490 $ xerbla
491* ..
492* .. External Functions ..
493 REAL SLAMCH
494 EXTERNAL slamch
495* ..
496* .. Intrinsic Functions ..
497 INTRINSIC max, sign, sqrt
498* ..
499* .. Executable Statements ..
500*
501* Decode and test the input parameters
502*
503 info = 0
504 lquery = ( lwork.EQ.-1 .OR. liwork.EQ.-1 )
505*
506 IF( ijob.LT.0 .OR. ijob.GT.5 ) THEN
507 info = -1
508 ELSE IF( n.LT.0 ) THEN
509 info = -5
510 ELSE IF( lda.LT.max( 1, n ) ) THEN
511 info = -7
512 ELSE IF( ldb.LT.max( 1, n ) ) THEN
513 info = -9
514 ELSE IF( ldq.LT.1 .OR. ( wantq .AND. ldq.LT.n ) ) THEN
515 info = -14
516 ELSE IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) ) THEN
517 info = -16
518 END IF
519*
520 IF( info.NE.0 ) THEN
521 CALL xerbla( 'STGSEN', -info )
522 RETURN
523 END IF
524*
525* Get machine constants
526*
527 eps = slamch( 'P' )
528 smlnum = slamch( 'S' ) / eps
529 ierr = 0
530*
531 wantp = ijob.EQ.1 .OR. ijob.GE.4
532 wantd1 = ijob.EQ.2 .OR. ijob.EQ.4
533 wantd2 = ijob.EQ.3 .OR. ijob.EQ.5
534 wantd = wantd1 .OR. wantd2
535*
536* Set M to the dimension of the specified pair of deflating
537* subspaces.
538*
539 m = 0
540 pair = .false.
541 IF( .NOT.lquery .OR. ijob.NE.0 ) THEN
542 DO 10 k = 1, n
543 IF( pair ) THEN
544 pair = .false.
545 ELSE
546 IF( k.LT.n ) THEN
547 IF( a( k+1, k ).EQ.zero ) THEN
548 IF( SELECT( k ) )
549 $ m = m + 1
550 ELSE
551 pair = .true.
552 IF( SELECT( k ) .OR. SELECT( k+1 ) )
553 $ m = m + 2
554 END IF
555 ELSE
556 IF( SELECT( n ) )
557 $ m = m + 1
558 END IF
559 END IF
560 10 CONTINUE
561 END IF
562*
563 IF( ijob.EQ.1 .OR. ijob.EQ.2 .OR. ijob.EQ.4 ) THEN
564 lwmin = max( 1, 4*n+16, 2*m*(n-m) )
565 liwmin = max( 1, n+6 )
566 ELSE IF( ijob.EQ.3 .OR. ijob.EQ.5 ) THEN
567 lwmin = max( 1, 4*n+16, 4*m*(n-m) )
568 liwmin = max( 1, 2*m*(n-m), n+6 )
569 ELSE
570 lwmin = max( 1, 4*n+16 )
571 liwmin = 1
572 END IF
573*
574 work( 1 ) = lwmin
575 iwork( 1 ) = liwmin
576*
577 IF( lwork.LT.lwmin .AND. .NOT.lquery ) THEN
578 info = -22
579 ELSE IF( liwork.LT.liwmin .AND. .NOT.lquery ) THEN
580 info = -24
581 END IF
582*
583 IF( info.NE.0 ) THEN
584 CALL xerbla( 'STGSEN', -info )
585 RETURN
586 ELSE IF( lquery ) THEN
587 RETURN
588 END IF
589*
590* Quick return if possible.
591*
592 IF( m.EQ.n .OR. m.EQ.0 ) THEN
593 IF( wantp ) THEN
594 pl = one
595 pr = one
596 END IF
597 IF( wantd ) THEN
598 dscale = zero
599 dsum = one
600 DO 20 i = 1, n
601 CALL slassq( n, a( 1, i ), 1, dscale, dsum )
602 CALL slassq( n, b( 1, i ), 1, dscale, dsum )
603 20 CONTINUE
604 dif( 1 ) = dscale*sqrt( dsum )
605 dif( 2 ) = dif( 1 )
606 END IF
607 GO TO 60
608 END IF
609*
610* Collect the selected blocks at the top-left corner of (A, B).
611*
612 ks = 0
613 pair = .false.
614 DO 30 k = 1, n
615 IF( pair ) THEN
616 pair = .false.
617 ELSE
618*
619 swap = SELECT( k )
620 IF( k.LT.n ) THEN
621 IF( a( k+1, k ).NE.zero ) THEN
622 pair = .true.
623 swap = swap .OR. SELECT( k+1 )
624 END IF
625 END IF
626*
627 IF( swap ) THEN
628 ks = ks + 1
629*
630* Swap the K-th block to position KS.
631* Perform the reordering of diagonal blocks in (A, B)
632* by orthogonal transformation matrices and update
633* Q and Z accordingly (if requested):
634*
635 kk = k
636 IF( k.NE.ks )
637 $ CALL stgexc( wantq, wantz, n, a, lda, b, ldb, q, ldq,
638 $ z, ldz, kk, ks, work, lwork, ierr )
639*
640 IF( ierr.GT.0 ) THEN
641*
642* Swap is rejected: exit.
643*
644 info = 1
645 IF( wantp ) THEN
646 pl = zero
647 pr = zero
648 END IF
649 IF( wantd ) THEN
650 dif( 1 ) = zero
651 dif( 2 ) = zero
652 END IF
653 GO TO 60
654 END IF
655*
656 IF( pair )
657 $ ks = ks + 1
658 END IF
659 END IF
660 30 CONTINUE
661 IF( wantp ) THEN
662*
663* Solve generalized Sylvester equation for R and L
664* and compute PL and PR.
665*
666 n1 = m
667 n2 = n - m
668 i = n1 + 1
669 ijb = 0
670 CALL slacpy( 'Full', n1, n2, a( 1, i ), lda, work, n1 )
671 CALL slacpy( 'Full', n1, n2, b( 1, i ), ldb, work( n1*n2+1 ),
672 $ n1 )
673 CALL stgsyl( 'N', ijb, n1, n2, a, lda, a( i, i ), lda, work,
674 $ n1, b, ldb, b( i, i ), ldb, work( n1*n2+1 ), n1,
675 $ dscale, dif( 1 ), work( n1*n2*2+1 ),
676 $ lwork-2*n1*n2, iwork, ierr )
677*
678* Estimate the reciprocal of norms of "projections" onto left
679* and right eigenspaces.
680*
681 rdscal = zero
682 dsum = one
683 CALL slassq( n1*n2, work, 1, rdscal, dsum )
684 pl = rdscal*sqrt( dsum )
685 IF( pl.EQ.zero ) THEN
686 pl = one
687 ELSE
688 pl = dscale / ( sqrt( dscale*dscale / pl+pl )*sqrt( pl ) )
689 END IF
690 rdscal = zero
691 dsum = one
692 CALL slassq( n1*n2, work( n1*n2+1 ), 1, rdscal, dsum )
693 pr = rdscal*sqrt( dsum )
694 IF( pr.EQ.zero ) THEN
695 pr = one
696 ELSE
697 pr = dscale / ( sqrt( dscale*dscale / pr+pr )*sqrt( pr ) )
698 END IF
699 END IF
700*
701 IF( wantd ) THEN
702*
703* Compute estimates of Difu and Difl.
704*
705 IF( wantd1 ) THEN
706 n1 = m
707 n2 = n - m
708 i = n1 + 1
709 ijb = idifjb
710*
711* Frobenius norm-based Difu-estimate.
712*
713 CALL stgsyl( 'N', ijb, n1, n2, a, lda, a( i, i ), lda, work,
714 $ n1, b, ldb, b( i, i ), ldb, work( n1*n2+1 ),
715 $ n1, dscale, dif( 1 ), work( 2*n1*n2+1 ),
716 $ lwork-2*n1*n2, iwork, ierr )
717*
718* Frobenius norm-based Difl-estimate.
719*
720 CALL stgsyl( 'N', ijb, n2, n1, a( i, i ), lda, a, lda, work,
721 $ n2, b( i, i ), ldb, b, ldb, work( n1*n2+1 ),
722 $ n2, dscale, dif( 2 ), work( 2*n1*n2+1 ),
723 $ lwork-2*n1*n2, iwork, ierr )
724 ELSE
725*
726*
727* Compute 1-norm-based estimates of Difu and Difl using
728* reversed communication with SLACN2. In each step a
729* generalized Sylvester equation or a transposed variant
730* is solved.
731*
732 kase = 0
733 n1 = m
734 n2 = n - m
735 i = n1 + 1
736 ijb = 0
737 mn2 = 2*n1*n2
738*
739* 1-norm-based estimate of Difu.
740*
741 40 CONTINUE
742 CALL slacn2( mn2, work( mn2+1 ), work, iwork, dif( 1 ),
743 $ kase, isave )
744 IF( kase.NE.0 ) THEN
745 IF( kase.EQ.1 ) THEN
746*
747* Solve generalized Sylvester equation.
748*
749 CALL stgsyl( 'N', ijb, n1, n2, a, lda, a( i, i ), lda,
750 $ work, n1, b, ldb, b( i, i ), ldb,
751 $ work( n1*n2+1 ), n1, dscale, dif( 1 ),
752 $ work( 2*n1*n2+1 ), lwork-2*n1*n2, iwork,
753 $ ierr )
754 ELSE
755*
756* Solve the transposed variant.
757*
758 CALL stgsyl( 'T', ijb, n1, n2, a, lda, a( i, i ), lda,
759 $ work, n1, b, ldb, b( i, i ), ldb,
760 $ work( n1*n2+1 ), n1, dscale, dif( 1 ),
761 $ work( 2*n1*n2+1 ), lwork-2*n1*n2, iwork,
762 $ ierr )
763 END IF
764 GO TO 40
765 END IF
766 dif( 1 ) = dscale / dif( 1 )
767*
768* 1-norm-based estimate of Difl.
769*
770 50 CONTINUE
771 CALL slacn2( mn2, work( mn2+1 ), work, iwork, dif( 2 ),
772 $ kase, isave )
773 IF( kase.NE.0 ) THEN
774 IF( kase.EQ.1 ) THEN
775*
776* Solve generalized Sylvester equation.
777*
778 CALL stgsyl( 'N', ijb, n2, n1, a( i, i ), lda, a, lda,
779 $ work, n2, b( i, i ), ldb, b, ldb,
780 $ work( n1*n2+1 ), n2, dscale, dif( 2 ),
781 $ work( 2*n1*n2+1 ), lwork-2*n1*n2, iwork,
782 $ ierr )
783 ELSE
784*
785* Solve the transposed variant.
786*
787 CALL stgsyl( 'T', ijb, n2, n1, a( i, i ), lda, a, lda,
788 $ work, n2, b( i, i ), ldb, b, ldb,
789 $ work( n1*n2+1 ), n2, dscale, dif( 2 ),
790 $ work( 2*n1*n2+1 ), lwork-2*n1*n2, iwork,
791 $ ierr )
792 END IF
793 GO TO 50
794 END IF
795 dif( 2 ) = dscale / dif( 2 )
796*
797 END IF
798 END IF
799*
800 60 CONTINUE
801*
802* Compute generalized eigenvalues of reordered pair (A, B) and
803* normalize the generalized Schur form.
804*
805 pair = .false.
806 DO 70 k = 1, n
807 IF( pair ) THEN
808 pair = .false.
809 ELSE
810*
811 IF( k.LT.n ) THEN
812 IF( a( k+1, k ).NE.zero ) THEN
813 pair = .true.
814 END IF
815 END IF
816*
817 IF( pair ) THEN
818*
819* Compute the eigenvalue(s) at position K.
820*
821 work( 1 ) = a( k, k )
822 work( 2 ) = a( k+1, k )
823 work( 3 ) = a( k, k+1 )
824 work( 4 ) = a( k+1, k+1 )
825 work( 5 ) = b( k, k )
826 work( 6 ) = b( k+1, k )
827 work( 7 ) = b( k, k+1 )
828 work( 8 ) = b( k+1, k+1 )
829 CALL slag2( work, 2, work( 5 ), 2, smlnum*eps, beta( k ),
830 $ beta( k+1 ), alphar( k ), alphar( k+1 ),
831 $ alphai( k ) )
832 alphai( k+1 ) = -alphai( k )
833*
834 ELSE
835*
836 IF( sign( one, b( k, k ) ).LT.zero ) THEN
837*
838* If B(K,K) is negative, make it positive
839*
840 DO 80 i = 1, n
841 a( k, i ) = -a( k, i )
842 b( k, i ) = -b( k, i )
843 IF( wantq ) q( i, k ) = -q( i, k )
844 80 CONTINUE
845 END IF
846*
847 alphar( k ) = a( k, k )
848 alphai( k ) = zero
849 beta( k ) = b( k, k )
850*
851 END IF
852 END IF
853 70 CONTINUE
854*
855 work( 1 ) = lwmin
856 iwork( 1 ) = liwmin
857*
858 RETURN
859*
860* End of STGSEN
861*
subroutine slassq(n, x, incx, scl, sumsq)
SLASSQ updates a sum of squares represented in scaled form.
Definition: slassq.f90:137
subroutine slacpy(UPLO, M, N, A, LDA, B, LDB)
SLACPY copies all or part of one two-dimensional array to another.
Definition: slacpy.f:103
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine stgexc(WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, LDZ, IFST, ILST, WORK, LWORK, INFO)
STGEXC
Definition: stgexc.f:220
subroutine slacn2(N, V, X, ISGN, EST, KASE, ISAVE)
SLACN2 estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vec...
Definition: slacn2.f:136
subroutine slag2(A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1, WR2, WI)
SLAG2 computes the eigenvalues of a 2-by-2 generalized eigenvalue problem, with scaling as necessary ...
Definition: slag2.f:156
subroutine stgsyl(TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, SCALE, DIF, WORK, LWORK, IWORK, INFO)
STGSYL
Definition: stgsyl.f:299
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
Here is the call graph for this function:
Here is the caller graph for this function: